Example Find the ratio of the maximum photon frequency in the Balmer series to the minimum frequency in the p aschen series in the spectrum of the hydrogen atom

The maximum photon frequency in the Balmer series corresponds to the transition from the n=2 energy level to the n=∞ energy level. The minimum frequency in the Paschen series corresponds to the transition from the n=3 energy level to the n=∞ energy level.<br /><br />Using the Rydberg formula, the frequency of a photon emitted during a transition between two energy levels in a hydrogen atom is given by:<br /><br />ν = R_H \* Z^2 \* (1/n1^2 - 1/n2^2)<br /><br />where ν is the frequency, R_H is the Rydberg constant, Z is the atomic number (for hydrogen, Z=1), n1 is the lower energy level, and n2 is the higher energy level.<br /><br />For the maximum photon frequency in the Balmer series (n1=2, n2=∞), the frequency is:<br /><br />ν_Balmer_max = R_H \* (1/2^2 - 0) = R_H / 4<br /><br />For the minimum frequency in the Paschen series (n1=3, n2=∞), the frequency is:<br /><br />ν_Paschen_min = R_H \* (1/3^2 - 0) = R_H / 9<br /><br />Therefore, the ratio of the maximum photon frequency in the Balmer series to the minimum frequency in the Paschen series is:<br /><br />ν_Balmer_max / ν_Paschen_min = (R_H / 4) / (R_H / 9) = 9/4<br /><br />So, the correct answer is 9/4.